The equation governing small amplitude waves on a string can be written as

$$\frac{\partial^{2} y}{\partial t^{2}}=\frac{\partial^{2} y}{\partial x^{2}}$$

The end points $x=0$ and $x=1$ are fixed at $y=0$. At $t=0$, the string is held stationary in the waveform,

$$y(x, 0)=x(1-x) \quad \text { in } \quad 0 \leq x \leq 1 .$$

The string is then released. Find $y(x, t)$ in the subsequent motion.

Given that the energy

$$\int_{0}^{1}\left[\left(\frac{\partial y}{\partial t}\right)^{2}+\left(\frac{\partial y}{\partial x}\right)^{2}\right] d x$$

is constant in time, show that

$$\sum_{\substack{n \text { odd } \\ n \geqslant 1}} \frac{1}{n^{4}}=\frac{\pi^{4}}{96}$$